Professor Jean-Pierre Demailly gave a lecture titled “Hermitian-Yang-Mills approach to the conjecture of Griffiths on thepositivity of ample vector bundles” at AMSS on November 17, 2020. In his lecture, given a vector bundle of arbitrary rank with ample determinant line bundle on a projective manifold, they propose a new elliptic system of differential equations of Hermitian-Yang-Mills type for the curvature tensor. The system is designed so that solutions provide Hermitian metrics with positive curvature in the sense of Griffiths – and even in the dual Nakano sense. As a consequence, if an existence result could be obtained for every ample vector bundle, the Griffiths conjecture on the equivalence between ampleness and positivity of vector bundles would be settled. Another outcome of the approach is a new concept of volume for vector bundles.
Jean-Pierre Demailly (born 1957) is a French mathematician working in complex analysis and differential geometry. Demailly entered the école Normale Supérieure in 1975. He received his Ph.D. in 1982 under the direction of Henri Skoda at the Pierre and Marie Curie University. He became a professor at Université Grenoble Alpes in 1983. Demailly's prizes include the Grand Prix Mergier-Bourdeix from the French Academy of Sciences in 1994, the Simion Stoilow Prize from the Romanian Academy of Sciences in 2006, and the Stefan Bergman Prize from the American Mathematical Society in 2015. He became an academician of the French Academy of Sciences in 2007. He was an invited speaker at the International Congress of Mathematicians in 1994 and a plenary speaker in 2006.